About

Florian Frick is an Associate Professor in the Department of Mathematical Sciences at Carnegie Mellon University. Prior to joining CMU, he was an H.C. Wang Assistant Professor at Cornell University and completed his Ph.D. at TU Berlin. He has held visiting positions at MSRI in Berkeley during the Fall semester of 2017 and at Freie Universität Berlin during the 2021/22 academic year. His research develops geometric and topological methods to solve problems of both geometric nature and those beyond, by "geometrizing" problems that benefit from topological techniques, which are particularly effective in tracking global phenomena. Among the topological methods he develops are existence and nonexistence results for equivariant maps and embeddings of topological spaces, as well as fixed-point theorems. Beyond algebraic and geometric topology, he applies these tools to diverse problems including chromatic numbers of hypergraphs in combinatorics, fair division results in game theory and convex geometry, the large-scale structure of point sets in computational topology, and polytope theory. Within geometric topology, his work includes the theory of embeddings, manifold triangulations, inscribability problems, and metric geometry.

Research topics

• Computer Security
• Computer Science
• Distributed computing
• Computer network
• Engineering
• Embedded system

Selected publications

• A $\mathbb{Z}_2$-Topological Framework for Sign-rank Lower Bounds

  arXiv (Cornell University) · 2026-04-02

  preprintOpen access1st authorCorresponding

  We develop a topological framework for proving lower bounds on sign-rank via $\mathbb{Z}_2$-equivariant topology, and use it to resolve the sign-rank of the Gap Hamming Distance problem up to lower-order terms. For every (partial) sign matrix $A$, we associate a free $\mathbb{Z}_2$-simplicial complex $S(A)$ and show that sign-rank of $A$ is characterized by the linear analog of $\mathbb{Z}_2$-index of $S(A)$. As a consequence, the classical $\mathbb{Z}_2$-index of $S(A)$ lower bounds the sign-rank of $A$, which reduces sign-rank lower bounds to topological obstructions. This reduction allows us to use various tools from $\mathbb{Z}_2$-equivariant topology, particularly in regimes where classical lower-bound techniques break down. As the main application, we consider the Gap Hamming Distance function $\mathrm{GHD}_k^n$ (defined for $k < n/2$), which distinguishes pairs of strings in $\{0,1\}^n$ with Hamming distance at most $k$ from pairs with distance at least $n-k$. We prove an essentially tight lower bound and show that for any $k$, \[ \text{sign-rank}(\mathrm{GHD}_k^n) = (1-o_k(1)) 2k. \] where the $o_k(1)$ term is $O\left(\sqrt{\frac{\log k}{k}}\right)$. This improves on the previous lower bound of Hatami, Hosseini, and Meng (STOC 2023) who proved that sign-rank of $\mathrm{GHD}_k^n$ is at least $Ω(k/\log(n/k))$. A key technical ingredient is a new analysis of the $\mathbb{Z}_2$-coindex (which lower bounds $\mathbb{Z}_2$-index) of the Vietoris-Rips complex of the hypercube in the sparse regime which yields an essentially tight lower bound. Previously, no results were known in the sparse regime.

  PublisherDOI

• A $\mathbb{Z}_2$-Topological Framework for Sign-rank Lower Bounds

  ArXiv.org · 2026-04-02

  articleOpen access1st authorCorresponding

  We develop a topological framework for proving lower bounds on sign-rank via $\mathbb{Z}_2$-equivariant topology, and use it to resolve the sign-rank of the Gap Hamming Distance problem up to lower-order terms. For every (partial) sign matrix $A$, we associate a free $\mathbb{Z}_2$-simplicial complex $S(A)$ and show that sign-rank of $A$ is characterized by the linear analog of $\mathbb{Z}_2$-index of $S(A)$. As a consequence, the classical $\mathbb{Z}_2$-index of $S(A)$ lower bounds the sign-rank of $A$, which reduces sign-rank lower bounds to topological obstructions. This reduction allows us to use various tools from $\mathbb{Z}_2$-equivariant topology, particularly in regimes where classical lower-bound techniques break down. As the main application, we consider the Gap Hamming Distance function $\mathrm{GHD}_k^n$ (defined for $k < n/2$), which distinguishes pairs of strings in $\{0,1\}^n$ with Hamming distance at most $k$ from pairs with distance at least $n-k$. We prove an essentially tight lower bound and show that for any $k$, \[ \text{sign-rank}(\mathrm{GHD}_k^n) = (1-o_k(1)) 2k. \] where the $o_k(1)$ term is $O\left(\sqrt{\frac{\log k}{k}}\right)$. This improves on the previous lower bound of Hatami, Hosseini, and Meng (STOC 2023) who proved that sign-rank of $\mathrm{GHD}_k^n$ is at least $Ω(k/\log(n/k))$. A key technical ingredient is a new analysis of the $\mathbb{Z}_2$-coindex (which lower bounds $\mathbb{Z}_2$-index) of the Vietoris-Rips complex of the hypercube in the sparse regime which yields an essentially tight lower bound. Previously, no results were known in the sparse regime.

  PublisherOA PDF

• Roots of real-valued zero mean maps: Compositions of linear functionals and equivariant maps

  European Journal of Mathematics · 2025-12-17

  articleOpen access

  Abstract We develop a novel topological framework that yields results constraining the distribution of zeros of certain zero mean real-valued maps, namely those obtained from composing a fixed equivariant map with linear functionals. We use this framework to establish upper bounds for the topology of set systems in the domain where (multivariate) trigonometric polynomials do not change their sign, generalizing and, in certain regimes, strengthening results in the literature. Our results more generally contain restrictions on the distribution of zeros of Chebyshev spaces as special cases. Lastly, we apply this framework to derive existence results for efficient cubature rules for compositions of affine functionals and equivariant maps.

  PublisherOA PDFDOI

• Youden's demon is Sylvester's problem

  Mathematika · 2025-02-18 · 2 citations

  articleOpen access1st authorCorresponding

  Abstract If four people with Gaussian‐distributed heights stand at Gaussian positions on the plane, the probability that there are exactly two people whose height is above the average of the four is exactly the same as the probability that they stand in convex position; both probabilities are . We show that this is a special case of a more general phenomenon: The problem of determining the position of the mean among the order statistics of Gaussian random points on the real line (Youden's demon problem) is the same as a natural generalization of Sylvester's four point problem to Gaussian points in . Our main tool is the observation that the Gale dual of independent samples in itself can be taken to be a set of independent points (translated to have barycenter at the origin) when the distribution of the points is Gaussian.

  PublisherOA PDFDOI

• Inter-Stream Dependencies in Time-Sensitive Networking

  2025-03-11 · 1 citations

  articleOpen access

  Effective configuration of Time-Sensitive Networks is crucial for providing timeliness and reliability guarantees for real-time industrial applications, where many inter-dependent streams may co-exist. However, existing methods for specifying the characteristics of real-time streams to the Centralized Network Configuration element (CNC) do not allow the specification of stream dependencies. Consequently, the CNC may schedule the real-time streams in a sub-optimal way, leading to low number of real-time applications supported. To address this pressing issue, we present methods that allow to model and express such inter-stream dependencies and propose extensions to the user network interface (UNI) to convey these dependencies to the CNC, transforming it into an intent-based UNI. We define a novel metric to evaluate a specific network configuration regarding the latency budget available for scheduling the transmission of a stream. Using a set of numerical simulations, we show that when exploiting the knowledge of inter-stream dependencies, we can derive better network schedules, which results in significant increase in applications onboarding up to 2.2x and increased network utilization for real-time streams up to 3x.

  PublisherOA PDFDOI

• Quantifying discontinuity

  ArXiv.org · 2025-11-10

  preprintOpen access

  Given a compact space $X$ that does not admit an embedding (an injective continuous function) into $\mathbb{R}^d$, we study the ''degree'' of discontinuity that any injective function $X \to \mathbb{R}^d$ must have. To this end, we define a scale invariant modulus of discontinuity and obtain general lower bounds, thus obtaining quantified nonembeddability results of Haefliger--Weber type. Moreover, we establish analogous lower bounds for simplicial complexes that do not admit an almost $r$-embedding in $\mathbb{R}^d$, thus obtaining a quantified version of the topological Tverberg theorem.

  PublisherOA PDFDOI

• Topological methods in zero-sum Ramsey theory

  Forum of Mathematics Sigma · 2025-01-01

  articleOpen access1st authorCorresponding

  Abstract A landmark result of Erdős, Ginzburg, and Ziv (EGZ) states that any sequence of $2n-1$ elements in ${\mathbb {Z}}/n$ contains a zero-sum subsequence of length n . While algebraic techniques have predominated in deriving many deep generalizations of this theorem over the past sixty years, here we introduce topological approaches to zero-sum problems which have proven fruitful in other combinatorial contexts. Our main result is a topological criterion for determining when any ${\mathbb {Z}}/n$ -coloring of an n -uniform hypergraph contains a zero-sum hyperedge. In addition to applications for Kneser hypergraphs, for complete hypergraphs our methods recover Olson’s generalization of the EGZ theorem for arbitrary finite groups. Furthermore, we give a fractional generalization of the EGZ theorem with applications to balanced set families and provide a constrained EGZ theorem which imposes combinatorial restrictions on zero-sum sequences in the original result.

  PublisherOA PDFDOI

• Covering and labeling generalizations of the Borsuk-Ulam theorem

  ArXiv.org · 2025-09-08

  preprintOpen access1st authorCorresponding

  We prove multiple generalizations of Fan's combinatorial labeling result for sphere triangulations. This can be seen as a comprehensive extension of the Borsuk--Ulam theorem. In typical applications, the Borsuk--Ulam theorem gives complexity bounds in a suitable sense, whereas our extension additionally provides insight into the structure of objects satisfying the complexity bound. This structure is governed by order types of finite point sets in Euclidean space and more generally by the intersection combinatorics of faces under continuous maps from the simplex. We develop some of those applications for sphere coverings, Kneser-type colorings, Hall-type results for hypergraphs, and hyperplane mass partitions, among other consequences. We provide a new proof of the topological Hall theorem and extend it into a result that simultaneously generalizes hypergraph Hall theorems and topological lower bounds for chromatic numbers.

  PublisherOA PDFDOI

• Hausdorff vs Gromov–Hausdorff Distances

  Discrete & Computational Geometry · 2025-02-19 · 1 citations

  article
  PublisherDOI

• Enabling the Sensor-To-Cloud Vision Through IT/OT Convergence – Industrial Power Consumption Measurement Device as a Reference Use Case

  Lecture notes in production engineering · 2025-01-01

  book-chapter
  PublisherDOI

Recent grants

• Topological Methods for Discrete Problems

  NSF · $180k · 2019–2023

Frequent coauthors

• Zoe Wellner

  21 shared

• Ling Hei Tsang

  The Ohio State University

  19 shared

• Megumi Asada

  Williams College

  19 shared

• Maxwell Polevy

  Cornell University

  19 shared

• David Stoner

  Stanford University

  19 shared

• Anne Shiu

  Texas A&M University

  17 shared

• Aaron Chen

  17 shared

• Vivek Pisharody

  17 shared

Education

• Ph.D.

  Technische Universität Berlin and Berlin Mathematical School

Awards & honors

• NSF CAREER Award